A303362 -id:A303362 - cp's OEIS frontend

A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are Heinz numbers of stable intersecting partitions.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Intersection of A316476 and A328867.
Heinz numbers of the partitions counted by A328871.
Replacing "intersecting" with "relatively prime" gives A328677.
Cf. A056239, A112798, A285573, A289509, A303362, A304713, A327393, A328671.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],stableQ[Union[primeMS[#]],GCD[#1,#2]==1&]&&stableQ[Union[primeMS[#]],Divisible]&]

A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 2, 7, 5, 7, 2, 10, 2, 11, 7, 14, 2, 16, 4, 19, 8, 22, 2, 30, 3, 29, 14, 37, 8, 48, 4, 50, 19, 59, 5, 82, 4, 81, 28, 93, 8, 128, 9, 128, 38, 147, 8, 199, 19, 196, 52, 223, 12, 308

Offset: 0

Gus Wiseman, Nov 12 2019

nonn

A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are just stable intersecting partitions.

			The a(1) = 1 through a(10) = 5 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      111111111  64
                           111111           11111111             22222
                                                                 1111111111

The Heinz numbers of these partitions are A329366.
Replacing "intersecting" with "relatively prime" gives A328676.
Stable partitions are A305148.
Intersecting partitions are A328673.
Cf. A000837, A285573, A303362, A305148, A316476, A328671, A328677.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[Union[#],Divisible]&&stableQ[Union[#],GCD[#1,#2]==1&]&]],{n,0,30}]

A305080 Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 2, 2, 3, 2, 2, 4, 2, 3, 4, 4, 3, 4, 3, 4, 5, 6, 4, 6, 5, 7, 6, 5, 6, 8, 6, 6, 6, 10, 11, 11, 9, 11, 9, 13

Offset: 1

Gus Wiseman, May 25 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

Conjecture: This sequence is "eventually increasing," meaning that for all k >= 0 there exists an m >= 0 such that a(n) > k for all n > m. For k = 0 it appears we can take m = 18, for example.

			The a(52) = 6 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,10,6): {{2,4},{2,3},{1,3},{1,2}}
(22,14,10,6): {{1,5},{1,4},{1,3},{1,2}}
     (30,22): {{1,2,3},{1,5}}
     (38,14): {{1,8},{1,4}}
     (42,10): {{1,2,4},{1,3}}
      (46,6): {{1,9},{1,2}}
The a(60) = 8 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
    (33,21,6): {{2,5},{2,4},{1,2}}
   (35,15,10): {{3,4},{2,3},{1,3}}
    (39,15,6): {{2,6},{2,3},{1,2}}
      (34,26): {{1,7},{1,6}}
      (38,22): {{1,8},{1,5}}
      (39,21): {{2,6},{2,4}}
      (46,14): {{1,9},{1,4}}

Cf. A006126, A048143, A087188, A302242, A303362, A303364, A304714, A304716, A305078, A305079.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@SquareFreeQ/@#,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]

A305761 Nonprime Heinz numbers of z-trees.

Original entry on oeis.org

91, 203, 247, 299, 301, 377, 427, 551, 553, 559, 611, 689, 703, 707, 791, 817, 851, 923, 949, 973, 1027, 1073, 1081, 1141, 1159, 1247, 1267, 1313, 1339, 1349, 1363, 1391, 1393, 1501, 1537, 1591, 1603, 1679, 1703, 1739, 1757, 1769, 1781, 1807, 1897, 1919, 1961

Offset: 1

Gus Wiseman, Jun 10 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph. The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)). Finally, a z-tree of weight n is a connected strict integer partition of n with at least two pairwise indivisible parts and z-density -1.

			2639 is the Heinz number of {4,6,10}, a z-tree corresponding to the multiset system {{1,1},{1,2},{1,3}}.

Cf. A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079, A305081.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Select[Range[3000],With[{p=primeMS[#]},And[UnsameQ@@p,Length[p]>1,zensity[p]==-1,Length[zsm[p]]==1,Select[Tuples[p,2],UnsameQ@@#&&Divisible@@#&]=={}]]&]

A327520 Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 5, 1, 1, 1, 2, 3, 1, 1, 7, 2, 2, 1, 1, 1, 4, 1, 2, 2, 1, 2, 1, 1, 11, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 15, 1, 7, 1, 1, 2, 2, 2, 1, 1, 4, 2, 1, 2, 1, 5, 1, 2, 1, 4, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476.

			The a(26) = 4 factorizations of 45 into stable numbers:
  (3*3*5)
  (3*15)
  (5*9)
  (45)
The a(201) = 11 multiset partitions of the prime indices of 495 into stable multisets:
  {{2},{2},{3},{5}}
  {{2},{2},{3,5}}
  {{2},{3},{2,5}}
  {{2},{5},{2,3}}
  {{2},{2,3,5}}
  {{3},{2,2},{5}}
  {{3},{2,2,5}}
  {{2,2},{3,5}}
  {{5},{2,2,3}}
  {{2,3},{2,5}}
  {{2,2,3,5}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A001055, A303362, A305149.

nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
y=Select[Range[nn],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

cp's OEIS Frontend

A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

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A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

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A305080 Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.

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A305761 Nonprime Heinz numbers of z-trees.

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A327520 Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.

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